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Heron's principle
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(Theorem)
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Theorem. Let $A$ and $B$ be two points and $l$ a line of the Euclidean plane. If $X$ is a point of $l$ such that the sum $AX\!+\!XB$ is the least possible, then the lines $AX$ and $BX$ form equal angles with the line $l$ .
This Heron's principle, concerning the reflection of light, is a special case of Fermat's principle in optics.
Proof. If $A$ and $B$ are on different sides of $l$ , then $X$ must be on the line $AB$ , and the assertion is trivial since the vertical angles are equal. Thus, let the points $A$ and $B$ be on the same side of $l$ . Denote by $P$ and $Q$ the points of the line $l$ where the normals of $l$ set through $A$ and $B$
intersect $l$ , respectively. Let $C$ be the intersection point of the lines $AQ$ and $BP$ . Then, $X$ is the point of $l$ where the normal line of $l$ set through $C$ intersects $l$ .
Justification: From two pairs of similar right triangles we get the proportion equations $$AP:CX \;=\; PQ:XQ, \quad BQ:CX \;=\; PQ:PX,$$ which imply the equation $$AP:PX \;=\; BQ:XQ.$$ From this we can infer that also $$\Delta AXP \sim \Delta BXQ.$$ Thus the corresponding angles $AXP$ and $BXQ$ are equal.
We still state that the route $AXB$ is the shortest. If $X_1$ is another point of the line $l$ , then $AX_1\,=\,A'X_1$ , and thus we obtain $$AX_1B \;=\; A'X_1B \;=\; A'X_1+X_1B \;\geqq\; A'B \;=\; A'XB \;=\; AXB.$$
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- TERO HARJU: Geometria. Lyhyt kurssi. Matematiikan laitos. Turun yliopisto, Turku (2007).
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"Heron's principle" is owned by pahio.
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Cross-references: equation, imply, proportion equations, right triangles, similar, normal line, intersect, normals, vertical angles, proof, reflection, angles, sum, Euclidean plane, line, points, theorem
There is 1 reference to this entry.
This is version 5 of Heron's principle, born on 2009-02-13, modified 2009-02-17.
Object id is 11619, canonical name is HeronsPrinciple.
Accessed 418 times total.
Classification:
| AMS MSC: | 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries) |
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Pending Errata and Addenda
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{$C$} \rput(0.44,-0.25){$X$} \rput(2.8,0){$l$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/11619/js/img1.png "\begin{pspicture}(-3,-1)(3,3) \psline(-2.6,0)(2.6,0) \psdots[linecolor=blue](-2,... ...$} \rput(0.44,1.3){$C$} \rput(0.44,-0.25){$X$} \rput(2.8,0){$l$} \end{pspicture}")
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